Statistical Hydrology

1. Statistical Hydrology • Read Chapter 2 (McCuen 2004) for background review • Supplementary materials: • Parameter Estimation: (a) Method of Moments * Product Moments (covered in CIVL181) * L-Moments (b) Method of Likelihood (covered in CIVL181) Statistical hydrology

2. Statistical Moments • Product Moments: E[Xr] - Disadvantages: (a) Sensitive to the presence of outliers; (b) Accuracy deteriorates rapidly as the order of moment increases • Probability-Weighted Moments (PWM): - Def: Mr,p,q = E{Xr [F(X)]p [1-F(X)]q } - Especially attractive when the CDF, F(x), has a closed- form expression - Special cases (a) Mr,0,0= E[Xr] (b) M1,0,q = aq ; M1,p,0 = bp Statistical hydrology

3. Relations between moments and parameters of selected distribution models (Tung et al. 2006) Statistical hydrology

4. Statistical Moments (2) • L-Moments: A linear combination of order statistics • Specifically, for the first 4 L-moments: Statistical hydrology

5. Graphical Representation of L-Moments Statistical hydrology

6. Analogy Between L- and Product-Moments Product Moments L-Moments m (mean) l1 (mean) s (stdev) l2 (L-std) Cv = s/mt2 = l2/l1 (L-Cv) Cs = m3/s3t3 = l3/l2 (L-Cs), | t3|<1 Ck = m4/s4t4 = l4/l2 (L-Ck), -0.25<t4<1 Statistical hydrology

7. L-moments & Distribution Parameter Relations From “Frequency Analysis of Extreme Events,” Chapter 8 in Handbook of Hydrology, By Stedinger, Vogel, and Foufoula-Georgiou, McGraw-Hill Book Company, New York, 1993 Statistical hydrology

8. Generalized Logistic Distribution Statistical hydrology

9. L-Moment Ratio Diagram Statistical hydrology

10. Statistical Moments (3) • Relations between L-moments and b-moments: Statistical hydrology

11. Sample Estimates of Statistical Moments Product Moments L- Moments Statistical hydrology

12. Example-1(a) Statistical hydrology

13. Example-1(b) Statistical hydrology

14. Example-1(c) Statistical hydrology

15. Types of Hydrologic Data Series Statistical hydrology

16. Return Period (Recurrent Interval) • The return period of an event is the time between occurrences of the events. The events can be those whose magnitude exceeds or equals to a certain magnitude of interest, i.e, XxT • In general, the actual return period (or inter-arrival time) between the occurrences of an event could vary. The ‘return period’ commonly used in engineering is the expected (or long-term averaged) inter-arrival time between events. • Return period depends on the time scale of the data. E.g., using annual maximum (or min.) series, the return period is year. • Return period T = 1/Pr[XxT] • To avoid misconception and mis-interpretation of an event, e.g., 50-year flood, it is advisable to use “flood event with 1-in-50 chance being exceeded annually”. Statistical hydrology

17. Distributions Commonly Used in Hydrologic Frequency Analysis • Normal Family – Normal, Log-normal • Gamma Family – Pearson type III, Log-Pearson type III • Extreme Value – Type I (for max. or min) - Gumbel Type II (for min) – Weibull Generalized Extreme Value Statistical hydrology

18. Graphical Frequency Analysis • Data are arranged in ascending order of magnitude, x(n) x(n-1) ··· x(2) x(1) • Compute the plotting position for each observed data Weibull plotting position formula: P[X≤x(m)]=m/(n+1) See other formulas • Plot x(m) vs. m/(n+1) on a suitable probability paper. (Commercially available are normal, log-normal, and Gumbel probability papers) • Extrapolate or interpolate frequency curve graphically. Statistical hydrology

19. Plotting Position Formulas Statistical hydrology

20. Example-2 (Graphical Procedure) Statistical hydrology

21. Normal Probability Plot Statistical hydrology

22. Log-normal Probability Plot Statistical hydrology

23. Gumbel Probability Plot Statistical hydrology

24. Frequency Factor Method Statistical hydrology

25. Frequency Factor for Various Dist’ns (1) Statistical hydrology

26. Frequency Factor for Various Dist’ns (2) Statistical hydrology

27. KT for Log-Pearson III Distribution Statistical hydrology

28. Analytical Frequency Analysis Procedure Statistical hydrology

29. Issues in Frequency Analysis • Selection of distribution and parameter estimation • Treatment of zero flows • Detection and treatment of outliers (high or low) • Regional frequency analysis • Use of historical and paleo data Statistical hydrology

30. Example (Analytical Procedure) Statistical hydrology

31. Confidence of Derived Frequency Relation Statistical hydrology

32. Uncertainty of Sample Quantiles Statistical hydrology

33. Approaches to Construct Confidence Interval Statistical hydrology

34. Standard Error of Sample Quantiles Statistical hydrology

35. (1-a)% CI for Sample Quantiles Statistical hydrology

36. Example (C.I.) Statistical hydrology

37. Hydrologic Risk • For a T-year event, P(XxT)=1/T. If xT is determined from an annual maximum series, 1/T is the probability of exceedance for the hydrologic event in any one year. • Assume independence of occurrence of events and the hydraulic structure is design for an event of T-year return period. Failure probability over an n-year service period, pf, is pf = 1-(1-1/T)n (using Binomial distribution) or pf = 1-exp(-n/T) (using Poisson distribution) • Types of problem: (a) Given T, n, find pf (b) Specify pf & T, find n (c) Specify pf & n, find T Statistical hydrology